Where does the wave function of the universe live?
Please describe its home.

I think this is the Hilbert space of the universe. (Greater or lesser, depending on which church you belong to.)
Or maybe it is the Fock space of the universe, or some still bigger, yet more complicated stringy thingy.

I will leave it to you whether you want to describe the observable universe, the total universe, or even the multiverse.

Please give a reasonably accurate and succinct mathematical description, including at least dimensionality.

I dont know why this is but the funny formulation of this question makes me chuckle somehow ;-). Stephen Hawking writes a bit about this issue in his "Great Design" book. He considers some kind of a path integral over the whole multiverse (each path corresponds to the evolution of one of the 10^**** individual members). The evolution of our own universe from the beginning to the present would then be described by the most probable (or classical limit) path.
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DilatonApr 10 '12 at 13:21

@Dilaton: I guess the book was grand design..
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Vineet MenonApr 10 '12 at 15:27

@VineetMenon Whoops yes, you are right. The typo is probably due to the fact that I thinkd this design is great, ha ha :-)
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DilatonApr 10 '12 at 15:32

I don't think there is any consensus about what a wavefunction of the universe means, let alone how to formulate it. Maybe someone in theoreticalphysics.stackexchange.com could comment.
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John RennieApr 10 '12 at 17:55

2 Answers
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More seriously, a wave function is a more special name of the "state vector" which is the element of the Hilbert space ${\mathcal H}$, a complex vector space with an inner product. The Hilbert space of all realistic systems is infinite-dimensional; for an infinite dimension, one can't really say whether the basis is countable or as large as a continuum because these two bases are actually fully equivalent.

Finite-dimensional Hilbert spaces are only used as simplified toy models for some aspects of some physical systems. But they're still very important in theory and practice because realistic situations are often composed of similar small Hilbert spaces by taking tensor products. The two-dimensional Hilbert spaces (e.g. spin-up vs spin-down) seem very simple but they're already very rich and are used as tools to teach quantum mechanics. Quantum computing usually takes place in Hilbert spaces for $N$ qubits which is $2^N$-dimensional, also finite-dimensional. The remaining infinitely many states of a real physical system are assumed to be inaccessible so we may "truncate" the Hilbert space. But note that systems as simple as en electron orbiting a proton or a harmonic oscillator already have an infinite-dimensional Hilbert space.

The Fock space is a special kind of Hilbert space. It is the Hilbert space of a free field theory or, equivalently, an infinite-dimensional harmonic oscillator. One usually defines the free - bilinear - Hamiltonian on the Fock space, too. If we don't say that there's a Hamiltonian, the identity of the Fock space is actually meaningless because all infinite-dimensional Hilbert spaces are isomorphic or "unitary equivalent" to each other.

So the Fock space isn't really "something completely different" (or larger) than the Hilbert space; it's a special case of it. The same thing holds for the Hilbert spaces associated with any theory you can think of (describing the world around us or describing a fictitious or hypothetical world), whether it's the Standard Model, the Minimal Supersymmetric Standard Model, or – the most comprehensive theory – String Theory. All these theories, much like any other theories respecting the postulates of quantum mechanics, have their own Hilbert space and all these infinite-dimensional spaces in string theory or a simple infinite-dimensional harmonic oscillator or even a simple Hydrogen atom are actually isomorphic to each other. The theories only differ by different Hamiltonians – or other dynamical laws that describe the evolution in time.

Also, one should mention that the actual state of the physical system isn't given by all the information included in an element of the Hilbert space. The phase and the absolute normalization – i.e. the full multiplicative factor that may be complex – is unphysical. So the space of inequivalent "pure states" is actually the quotient ${\mathcal H}/{\mathbb C}^*$.

Aside from "wave functions" i.e. pure states that are elements of the Hilbert space, up to a normalization, one may also describe a physical system by a more general "density matrix" which lives in the space of Hermitian matrices $\rho$. For pure states, $\rho=|\psi\rangle\langle\psi|$ and the phase cancels. However, there are also more general mixed states that are superpositions of similar terms.

I am having a lot of trouble with this one universal infinite dimensional Hilbert space. if the dimensionality of the universe doesn't tell us, how do we know if there is one spatial dimension or three or seventy seven? Also how do we know how many particles are in our universe? TIA
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Jim GraberApr 11 '12 at 12:25

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The Hilbert space is a mathematical construction that has essentially nothing to do with real space. So saying that the Hilbert space is infinite-dimensional does not imply anything about the spatial dimensionality of the universe. That is a separate issue.
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David Z♦Apr 11 '12 at 19:40

Dear @Jim, I agree with David. In QFT, you may try to determine the spacetime dimensions by isolating one-particle states and finding that the Hilbert space of one-particle states has a simple basis diffeomorphic to $R^{d-1} $, the spatial momentum, or in an equivalent way. But the Hilbert space is much greater than the regular space. Every basis vector of the Hilbert space in a basis corresponds to one mutually exclusive state in which the whole physical system may be. There are usually infinitely many.
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Luboš MotlApr 12 '12 at 12:17

So the choice of a greater or lesser Hilbert space only makes a difference for finite-dimensional Hilbert spaces? Because for an infinite Hilbert space the two are the same? Or at least isomorphic?
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Jim GraberApr 14 '12 at 10:33

The youtube video you linked to is dead. Maybe the wave function is vacationing at the Hilbert Hotel? .... I'll see myself out
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David HSep 7 '13 at 3:54

The word "space" in mathematics is not the same ontological object as physical space. It is sort of equivalent to asking, in classical physics, where is the space of all velocity vectors located? Its not that they are actually somewhere "out there", they are just mathematical abstractions from which useful information can be extracted and inferences can be made unto measurable quantities which are analogous to the abstraction. A vector space (Hilbert, Fock, and many other variations of vector spaces exist!) is a mathematical object which makes convenient many computations one can do with sets of numbers(vectors, matrices, tensors, etc...) endowed with human-invented algebraic properties (closure, commutativity, associativity, etc...). Quantum mechanics makes use of linear algebra almost solely for the fact that one can extract 'spectrum' of eigenvalues which are analogous to the measurable discrete quantities one finds when dealing with such objects.